On the characteristic and diameter of planar integral point sets

TL;DR

This paper improves the lower bounds on the diameter of planar integral point sets with specific characteristics, using hyperbola and grid-based methods to advance understanding of their geometric properties.

Contribution

It introduces new lower bounds for the diameter of integral point sets with characteristics of the form 4k+1 or 4k+2, combining hyperbola and grid techniques.

Findings

Enhanced lower bounds for the diameter of certain integral point sets

Application of hyperbola-based methods in geometric bounds

Use of grid-based approaches to improve bounds

Abstract

A point set $M$ in Euclidean plane is called an integral point set in semi-general position if all the distances between the elements of $M$ are integers, and $M$ does not contain collinear triples. We improve the lower bound for diameter of such sets in the particular case when the characteristic of the set is of the form $4k+1$ or $4k+2$. To achieve that, we combine hyperbolae-based and grid-based toolsets.